a) If r is the radius of a circle then:

\displaystyle \text{ Perimeter } = 2 \pi r

\displaystyle \text{Area   } = \pi r^2  \text{ or  Area }  = \frac{1}{4} \pi d^2

\displaystyle \text{Diameter   } (d) = 2r

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b) If r is the radius of a circle then:

\displaystyle \text{Perimeter of a semi circle   } = \pi r + 2r = (\pi + 2)r

\displaystyle \text{Area of semi circle   } = \frac{1}{2} \pi r^2

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c) If r is the radius of a circle then:

\displaystyle \text{Perimeter of the quadrant   } = \frac{1}{4} \times 2 \pi r + 2 r = ( \frac{\pi}{2}  + 2)r

\displaystyle \text{Area of the quadrant   } = \frac{1}{4} \pi r^2

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d) If R and r are radii of two concentric circles, then

\displaystyle \text{ Area enclosed between the two circle } \\ \\ = \pi R^2 - \pi r^2 = \pi (R^2 - r^2) = \pi (R+r)(R-r)

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2019-01-06_9-12-44e) If  \triangle ABC is an equilateral triangle of side \displaystyle a \text{ and height } h \text{ , then }  h = \frac{\sqrt{3}}{2} a .

Also if R and r are the radii of the circumscribed and inscribed circles of \displaystyle \triangle ABC , then \displaystyle R = \frac{2h}{3} \text{ and } r= \frac{h}{3}

\displaystyle \text{Circumference of the circumcircle of    } \triangle ABC = 2\pi R = 2 \pi \times  \frac{2h}{3} + \frac{4}{3}  \pi h

\displaystyle \text{Area of the circumcircle   } = \pi R^2 = \pi (  \frac{2h}{3}  )^2 =  \frac{4}{9}  \pi h^2

\displaystyle \text{Circumference of the inscribed circle   } = 2 \pi r =  \frac{2 \pi}{3}  h

\displaystyle \text{Area of the inscribed circle   } = \pi r^2 =  \frac{1}{9}  \pi h^2

f) Important notes:

  • If two circles touch internally, then the distance between their centers is equal to the difference of their radii.
  • If two circles touch externally. Then the distance between their centers is equal to the sum of their radii.
  • The distance moved by a rotating wheel in one revolution is equal to the circumference of the wheel.
  • The number of revolutions completed by a rotating wheel in one minute \displaystyle = \frac{\text{Distance moved in one minute}}{\text{circumference}}

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